_____________________________________________________________________________________________________ *Corresponding author: Email: eytansuchard@gmail.com, eytan_il@netvision.net.il;

Electro-Gravity Via Geometric Chronon Field

Eytan H. Suchard1*

ORIGINAL PAPER AT: PHYSICAL SCIENCE INTERNATIONAL JOURNAL DOI: 10.9734/PSIJ/2015/18291 http://sciencedomain.org/abstract/9858 THANKS TO THE PROFESSIONAL PEER REVIEW PROCESS THAT HAD SIGNIFICANTLY IMPROVED THE PAPER. INCLUDED: SMALL CORRECTION AND REMARK ON THE MAJORANA CHARGELESS FIELD ABSTRACT

Aim: To develop a model of matter that will account for electro-gravity. Interacting particles with non-gravitational fields, can be seen as clocks whose trajectory is not Minkowsky geodesic. A field in which a small enough clock is not geodesic, can be described by a scalar field of time whose gradient has non-zero curvature. The scalar field is either real which describes acceleration of neutral clocks made of charged matter or imaginary, which describes acceleration of clocks made of Majorana type matter. This way the scalar field adds information to space-time, which is not anticipated by the metric tensor alone. The scalar field cant be realized as a coordinate because it can be measured from a reference sub-manifold along different curves. In a Big Bang manifold, the field is simply an upper limit on measurable time by interacting clocks, backwards from each event to the big bang singularity as a limit only. In De Sitter / Anti De Sitter space-time, reference sub-manifolds from which such time is measured along integral curves, are described as all events in which the scalar field is zero. The solution need not be unique but the representation of the acceleration field by an anti-symmetric matrix, is unique up to SU(2) x U(1) degrees of freedom. Matter in Einstein Grossmann equation is replaced by the action of the acceleration field, i.e. by a geometric action which is not anticipated by the metric alone. This idea leads to a new formalism of matter that replaces the conventional stress-energy-momentum-tensor. The formalism will be mainly developed for classical but also for quantum physics. The result is that a positive charge manifests small attracting gravity and a stronger but small repelling acceleration field that repels even uncharged particles that measure proper time, i.e. have rest mass. Negative charge, manifests a repelling anti-gravity but also a stronger acceleration field that attracts even uncharged particles that measure proper time, i.e. have rest mass.

Keywords: Time; general relativity; electro-gravity; reeb class; godbillon-Vey class.

http://sciencedomain.org/abstract/9858

Suchard

1. INTRODUCTION The motivation of this theory is to show matter is an acceleration field i.e. prohibition of inertial motion/free-fall of every small particle that can measure proper time. The scalar field is either real which describes acceleration of neutral clocks made of charged matter or imaginary, or partially imaginary with some condition which describes acceleration of clocks made of Majorana type matter. Non-geodesic motion as a result of interaction with a field, is not a geodesic motion in a gravitational field, i.e. it is not free fall. Moreover, material fields by this interpretation prohibit geodesic motion curves of particles moving at speeds less than the speed of light and by this, reduce the measurement of proper time! The expression of such a field is thus by a field of time that is not part of the geometry of space-time but rather adds new information to the space-time manifold by expressing how material force fields bend the curve of a small enough clock. This bending is not anticipated by general relativity but can account for a complementary field to the gravitational field and it offers a new expression of matter. In this paper, the author replaces the notion of a force field by an acceleration field. In fact, the covariant description of such an acceleration field is by an anti-symmetric matrix that is multiplied by the velocity vector in order to point to a perpendicular direction i.e. to an acceleration 4-vector. A 4-vector of acceleration is only a representative of such a field, which as we shall see, leaves an SU(2)xU(1) degrees of freedom in the definition of the matrix. An acceleration can be seen as curvature of a particle's trajectory, however, we seek a formalism that will be independent of any specific trajectory. This goal can be achieved by an introduction of a very special scalar field, namely, a field of time. By the principles of General Relativity, no coordinate of time should be preferable and therefore any such scalar field should not lead to a realizable preferable coordinate of time. Also, because more than one curve can measure the same maximal proper time between a predefined reference sub-manifold and an event, a definition of maximal time may not lead to unique integral curves. The theory that will be presented is a geometric interpretation of Sam Vaknins Chronon Theory [1] by a Godbillon-Vey [2],[3] curvature vector / Reeb vector. This vector is a part of the Godbillon-Vey 3rd order form and it belongs to the De Rham Cohomology. Locally, it offers

foliations which their tangent bundle is the kernel of a 1-Form, namely the gradient of a scalar field of time. By the principle of parsimony, even the maximum time requirement is not necessary if the scalar field that defines proper time along curves, is emergent out of a mathematical formalism. Here we should introduce an equivalence relation. 1.1 Avoidance of Closed Time-like Curves In this paper, the curves along which maximal proper time is measured do not contain backward travel in time. If two events and

are on a curve such that is in the future

relative to then no geodesic exists between

and such that is in the future in

relation to , otherwise the curve is excluded from the calculation of maximal time.

1 e 2 e

2 e

1 e

1 e

2 e2 e 1 e

The equations of the presented theory may render this definition redundant. 1.2 Geodesic Equivalence between

Events Given a sub-manifold in 1M M . Events in a set of events are equivalent, if for each event

in there exists an event in such that the maximal proper time

2M

2M2 Me

Max

1 Me 1M

measured along curves that connect to events in is the same for each other event in

, i.e. If the maximal proper time from event

2 Me

1M

2M

2~

Me to is to event 1M 11 ~ MeM then the curve length in proper timer is also Max . The main interest of this paper is that for each Max , there exists a class such that e is a surjective map.

2MMax

1 M2 M e is subject to acceleration

fields, i.e. to fields that prevent any small test particle from moving along geodesic curves.

may not be unique and all such sub-manifolds of

1M

1MM should be dictated by a physical

law, i.e. a minimum action principle. Before we continue, we have to define the big bang as it embodies the most simple case of

. Please note that may not be a neat 1M 2M

2

Suchard

geometric object as a 3D sub-manifold. It can be a countable unification of such sub-manifolds. Big Bang: The Big Bang in this paper is a presumed event or manifold of events, such that if looking backwards from any clock - Test Particle, at an event e - that measures the maximum possible time up to e then such clock must have started the measurement from the big bang as a limit. The Big Bang synchronizes all possible such clocks that measure the maximal time to any event from the past. The definition allows prior time even before such a manifold of events but requires that clocks will be synchronized on the "big bang" manifold or that the synchronization will be done by measuring the limit of time backwards to a "big bang" singularity. The idea of a test particle measuring time and even transferring time is not new, thanks to Sam Vaknin's dissertation from 1982 in which he introduced the Chronon field [1] in an amendment to Dirac's equation. Instead of defining the maximal time backwards to a single event, the definition can be to a sub-manifold, but to which sub-manifold and what if small test particles are not allowed to move along geodesic curves due to interactions with force fields ? The physical law that we reach will have to solve that problem too. An earlier incomplete paper of the author about this inertial motion prohibition in material fields, can be found [4]. In general, the Euler number of the gradient of the time field may not be zero [5]. To avoid such singularities, this paper harnesses some of the results known to belong to Lars Hormander by using Distribution Theory [6]. If our time field is , we may consider a new scalar field P such that is complex. ** PP

* avoids

gradient field singularities where becomes zero if the derivatives of are discontinuous. If test particles are forced to move along non-geodesic curves, i.e. experience trajectory curvature due to an external field, a mathematical formalism of such curvature will have to be developed and will replace matter in Einstein-Grossmanns field equations, as such curvature fields become a new description of matter. It is quite known that acceleration can be seen as a curvature and therefore acceleration field is another interpretation of a curvature

vector perpendicular to 4-velocity [7] though it leaves SU(2)xU(1) degrees of freedom. An acceleration field that acts on any particle, can't be expressed as a 4-vector because a 4-vector does depend on a specific trajectory and by Tzvi Scarr and Yaakov Friedman, such a field is expressible by an anti-symmetric matrix

AA such that if is the normalized 4-velocity and then the 4-acceleration is

actually such that is the speed of light.

V

1VV

AC 2

a / V C

2. THE CLASSICAL NON-RELATIVISTIC

LIMIT MASS AT REST IN A GRAVITATIONAL FIELD

2.1 Definitions Gravity: Gravity is the phenomenon that causes all forms of energy to be inertial if and only if they freely fall, including a projectile that starts upwards. All forms of energy including light appear to accelerate towards the source of gravity. Gravity is seen as a phenomenon that influences the metrics of space-time. Mass Dependent Force: A mass dependent force is a presumed force that accelerates any massive object that does not propagate at the speed of light and the force is mass dependent. The mass dependent force does not change the metrics of space-time. i.e. clocks in the field will not tick slower than clocks far from the field unless gravity coincides. An Acceleration Field / Non-Inertial Field: An acceleration Field is Mass Dependent Force that is not intrinsic to an object but rather appears as a property of space-time. Unlike gravity, it does not affect photons or any particle that propagates at the speed of light. The idea is that such a field affects measurement of time by prohibition of inertial motion of particles that can measure proper time. Non geodesic curves in Minkowsky space-time measure less time between events. As was already mentioned, an acceleration field is expressible as and VAca

2/

AA is locally similar to the Tzvi Scarr and Yaakov Friedman matrix [7]. Very important: The acceleration field need not represent an interaction of a small test particle with the matter in which the field appears. As a

3

Suchard

property of space-time, it can represent an interaction with the entire space time. Caution: Although throughout the classical non-relativistic limit sections of this paper, the author uses vectors to describe trajectory dependent representative acceleration vectors, an acceleration field cant be described as a covariant field without the Scarr-Friedman [7] formalism and . As an author, it is important to me to make sure that the reader fully understands that classical limit calculations are nothing more than classical limit calculations and the author is fully aware of and does not mix non relativistic and covariant formalisms!

VAca

2/ 1VV

Motivation beyond this section: For the pedant physicist there is no point in presenting a potential intrinsic to a massive object and a non-relativistic potential energy as the classical limit of a covariant theory. To such a reader, the author will say that the purpose of this paper is to replace the conventional energy momentum tensor which is part of Einstein-Grossmanns field equation by a tensor with fully geometric meaning. Recall Einstein-Grossmanns field equations in his writing

convention as

T

RgRTC

K218

4 such

that K is the gravity constant, , the speed of light, the Ricci tensor, the metric

tensor, the Ricci scalar. The replacement will be with a totally geometric tensor and thus will achieve a gravity equation which is geometric on both sides. To give a further clue, the author will say that will be replaced by a tensor which is the result of a

representative acceleration

CgR

R

Rg

T

2Ca . 2C

a seems as

a curvature vector of a particles trajectory with units of 1/length but as such, it is an intrinsic property of the particle and not of a field. So eventually, we will have to derive our curvature vector from the gradient of a scalar field and not from the velocity of any specific particle. Since our new tensor is purely geometric, the constant

4

8C

K will be replaced by 1. To be more precise,

the equation will be written as 8 or another value e.g. 4 and

RgR

Ktermsotheraa

CK

21)

_(8 4

and in some special cases, where electric charge is not involved i.e. , a simple

equation will be valid,

0Q

Ra 4a

C

. In any

case this implies that Kaa

K

2a

can be

construed as energy density and hopefully the reader is not annoyed by the sloppy notation . 2a Using a potential field intrinsic to an object instead of correct covariant formalism and using gravitational pseudo-acceleration, the following will shed some light on the general intuition as to the expected relation between energy and acceleration fields although as a physical argument, it is not fully acceptable. We will now consider classical non-relativistic gravity and classical non-relativistic acceleration as qualitative limits that will hint at the relationship between non - inertial and non - geodesic acceleration fields and energy. Estimates will be discussed in the next section. The classical non-relativistic and non-inertial acceleration caused by material fields will be denoted by

),,( zyx aaaa . By the principle of equivalence, we will calculate the integral of the square acceleration of a particle at rest which is accelerated because it is prevented from inertial motion, i.e. from falling in a gravitational field of a ball of mass. We will see how this value is related to classical potential energy. The qualitative hint starts with integration. K is the constant of gravity, M mass, r radii of a hollowed ball quite the way it is done in the electromagnetic theory

4Kdr

0

0 0

22

2

22 4

r

VolumeV rKMr

rKMdVa (1)

Now we calculate the negative potential energy

gE ,

M

Er

KMdmr

Km

0 0

2

0 2g (2)

So from (1) and (2)

4

Suchard

gVolumeV

EdVaK

2

81

(3)

(3) qualitatively implies the following relation between energy and non-inertial acceleration where is the energy density and 2 C is the mass density

Ka

Ka

DensityEnergyCK

Ca

8421

_ 8

22

24

2

(4)

In special relativity, the square norm of a normalized by C , 4-velocity of a particle is constant and also 1

iiuuN

2

0)(

),(,2 ki

iiik dx

uuduuN k such that

Cddxu

ii and the normalized by , 4-

acceleration is

C

22

2

dCxdai

2

i

which is 1/length in

units which is the curvature of a specific particles trajectory. If was not the norm of a particles velocity, we could think of another way to describe acceleration. More or less, that will be the subject of more advanced sections of this paper.

N

3. THE CLASSICAL NON-RELATIVISTIC

LIMIT THE ELECTROSTATIC FIELD The following uses the standard definitions of electric and electrostatic fields. What can we say about the density of the electrostatic field? We know it is

20

2_ EDensityEnergy (5)

such that 0 is the permittivity of vacuum and E is the electrostatic field. (4) has a very deep meaning which is, that acceleration of neutral charge-less test particles should appear also within an electric field. That is because acceleration of all small enough rest mass due to the existence of a field, is assumed in this paper to be the reason of all energy densities including

the electric field. Prohibition on geodesic Minkowsky motion can be concentrated as an acceleration field in some areas around electric charge and may not be well distributed, however, such argument is less appealing if we consider the principle of parsimony.

EKaEK

a0

202

428

(6)

Definition: The constant that divides the non-geodesic square acceleration is called "Electro-gravity constant". In this paper we choose as an educated guess K8 . In general we can write our guess as 8 so (6) becomes:

EK

aEK

a22

0202

(7)

Other constants yield different theories and have dramatic cosmic consequences. (6) implies a very weak acceleration i.e. mass dependent force on small enough charge-less neutral test particles, about in a field of 1000000 volts over 1 millimeters distance. See Timir Datta et. al. work as an elegant way to focus field lines by metal cone and plane and to observe the effect [8], however, in this paper we shall see that there is another effect due to gravity and therefore the acceleration in (6) is not the only effect that has to be taken into account. The acceleration in (6) exposes non-inertial, non-gravitational acceleration of particles that can measure proper time. On its own, it is not an interesting acceleration but it can explain the electric interaction as repulsive when the integration of the square acceleration increases and attractive when this integration is reduced. The author believes the acceleration of charge-less particles in an electric field is from positive to negative.

2sec/61.8 cm

In Electro-gravitational thrust, Dark Matter and Dark Energy it will be shown that there is an electro-gravitational effect opposite in direction to the acceleration of an uncharged particle in an electro-static field. There is at least informal evidence that the elecro-gravitational effect shows thrust of the entire dipole towards the positive direction [9] and the author does not imply asymmetrical capacitors of 1 - 0.1 Pico-Farad with 45000 Volts. Such capacitors according to the calculations in the section Electro-gravitational thrust, Dark Matter and

5

Suchard

Dark Energy in this paper, cant manifest any measurable effect of at least 1 micro Newton thrust. Here is a testimony of Hector Luis Serrano, a former NASA physicist in reply to Peter Liddicoat: Actually by the generally accepted definition of what constitutes high vacuum 10^-6 Torr is about in the middle. This pressure is about equal to low Earth orbit. More importantly at this pressure the Mean Free Path of the molecules in the chamber is far too great to support Corona/Ion wind effects. Weve tested from atmosphere to 10^-7 Torr with no change in performance either. However, Im glad the results have you thinking. It looks simple, but trust me its not. 3.1 Serious Experimental Problem

Electron Mobility The down side of the non-geodesic acceleration is that it is about 10 orders of magnitude smaller than the accepted and known electric field interaction. For example, negative charge suspended above the Earth will cause charge to move in the ground. This charge will have a much stronger effect than the interaction with the acceleration field as is, and will cause a shielding effect i.e. the fields will cancel out within the Earth. Even the almost ideal insulator, i.e. diamond crystals, have impurities such as Nitrogen Vacancies [10] that allow charge carriers to move in the lattice i.e. high electron mobility. In the most pure diamonds the NV impurities are about nodes per comparing to carbon atoms per

. The donor electrons lie deep in the band gap of 5.47ev, at about 1.7 ev.

18102310

3cm77.1 x

3cm

4. THE NON-GEODESIC ACCELERATION

FIELD The authors strategy in developing the idea of an acceleration field that is not gravity is as follows:

1) The curvature vector of the gradient of a scalar field will be developed. If the meaning of the scalar field is that it is proper time, measured along different curves from a reference sub-manifold, then non-zero curvature vector means acceleration. The trajectory of any small enough clock that measures non-zero proper time will not be parallel to any

geodesic field as it interacts with material fields.

2) A specific 4-vector cant account for an acceleration field because we need a representation that does not depend on any specific velocity 4-vector. Such an acceleration is exactly Zvi Scarr and Yaakov Friedman matrix [7]. A basic singular matrix will be developed. It is singular because 2 vectors are not enough to describe rotation and scaling in Minkowsky space-time. There are still at least two more degrees of freedom and if our curvature vector is complex then there are SU(2)xU(1) degrees of freedom.

3) We develop a non-singular acceleration matrix in which there is SU(2)xU(1) degrees of freedom.

4) We represent the gradient of our scalar field of time by means of the perpendicular foliation and show an additional SU(3) degrees of freedom. Although SU(3)xSU(2)xU(1) is the symmetry group of the Standard Model, it is shown in this paper to be a result of geometry and not of any second quantization technique.

5) We explore more scaling degrees of freedom in the definition of the time field.

6) We use the square norm the second power of the Minkowsky norm of the curvature field as an action operator that can replace the material action in Einstein-Grossmann-Hilbert action.

7) We show that the Euler Lagrange equations lead to a new term that expresses the divergence of the curvature field. This divergence is attributed to electric charge because an electric field is a form of energy for which the simplest explanation is a very small acceleration of even uncharged small clocks along electro-static field lines. So electric interaction is predicted as either increasing or decreasing the energy of the acceleration field.

8) The interpretation of charge is of having two unpredicted fields, an acceleration field and an opposite weaker gravitational field. Electrons are predicted to manifest attractive acceleration field and a weaker repulsive gravity. Free intergalactic electrons are thus prime candidates for Dark Matter.

9) Warp drive is mentioned as a result of very large charge separation. Implementation by radio photons as possibly behaving as oscillating pairs of virtual +e and e charge

6

Suchard

is only briefly discussed though even slight imbalance of such virtual charge distribution is equivalent to large amounts of ordinary separated charge and may therefore account for Roger Shawyers EMDrive, Warp Drive, experiments by Dr. David Pares from Nebraska and for NASAs approval in the late news that EMDrive indeed involves Warp Drive effects. This idea also offers energy production promises to developing countries by Dennis Sciama inertial induction. This idea will be reminded.

10) The time field in Schwarzschild solution is discussed.

11) Ideas of how to extend this theory to quantum mechanics are offered by stochastic calculus.

It is first required to achieve a curvature field without resorting to Tzvi Scarr and Yaakov Friedman representation [7] that is required for a general acceleration field. such that

if is the 4-velocity such that

then

the 4-acceleration is actually

AA

VVV 1

VA2ca

. In

special relativity

22 /1

/,/,/,1

cv

cvcvcvV zyx

such that zyx ,, are the well known three dimensional Cartesian coordinates, are three dimensional velocity coordinates, the speed of light. The first coordinate is

zyx vvv ,,c

2c2 /1/1 v is the speed along the time axis. The vector field that this paper uses is the gradient of the scalar field of

time, dxdPPP , and it will replace the

velocity vector of specific particles. Please note that due to intersection of different curves,

*PPP

is not a velocity, however it is a unit

vector and 2*

U

PPPA should

emerge for some vector that replaces acceleration. The following is simply an exercise in differential geometry. Considering a scalar

field

U

P

and its gradient dxdPP in covariant

writing, such that are the coordinates, find the second power of the curvature of the field of

curves generated by

dx

dxdPP . It is a problem

in differential geometry that can be left for the reader as an exercise. However, if the reader wants to get the answer without too much effort along with some physical interpretations, he/she should read the following. The idea is to use a scalar field of time - that represents the maximum possible time measured by test particles - back to the big bang singularity as a limit or to a sub-manifold of events - and from this non -physical observable, to generate observable local measurements. The square curvature of a conserving vector field is defined by an arc length parameterization along the curves it forms.

t

Caution: This may not be the time measured by any physical particle because the scalar field from which the vector field is derived may be the result of an intersection of multiple trajectories. However, a particle that follows the gradient curves will indeed measure t even if its trajectory is not geodesic.

t

Let our time field be denoted by and let

denote the derivative by coordinates

P P

dxdP

P

or in Einstein-Grossmanns convention

P P, . Let t be the arc length measured along the curves formed by the vector field which may not be always geodesic due to intersections. By differential geometry, we know that the second power of curvature along these curves is simply

P

gPP

Pdtd

Pdtd

kk

kk

)()(2 P

PCurv (8)

such that is the metric tensor. For

convenience we will write

g

kk PPNorm and

PdtdP . For the arc length parameter t .

7

Suchard

8

Here it is the main trick, may not be constant because is NOT the 4-velocity of a specific particle, also but not only (see

APPENDIX The time field in the Schwarzschild solution), due to intersections of more than one possible particle trajectory curve.

NormP

Let denote: W

kk

kk

gPPNorm

P

PP

PdtdW 3)(

NormP

(9)

Obviously

03 NormgPP

NormgPPgPP

NormgPP

NormgPPgPW

kk

kkk

k

ss

kkk

k

(10)

Thus

22242

2 )(Norm

PPNorm

PPgPPNorm

gPPNorm

gPPWWCurv kks

s

(11)

Following the curves formed by dxdPPP , , The term

NormP

dtdxr is the derivative of the

normalized curve or normalized velocity, using the upper Christoffel symbols, s

rsrr PPdxdP ; .

Caution: Using normalized velocity, here has a differential geometry meaning but not a physical meaning because a physical particle will not necessarily follow the lines which are generated by the curves parallel to the gradient unless in vacuum. may result from an intersection of curves along which particles move but may not be parallel to any one of such curves intersecting with an event !!!

P P

NormPP

dtdxPP

dxdP

dtd r

r

rs

rsr );()( such that denotes the local coordinates. If is

a conserving field then and thus

rx P

;; rr PP ,21, 2NormPP rr and

)),

(,,

(41

)(

23

2

4

22

222

2

NormgPNorm

NormgNormNorm

NormPP

NormPPCurv

srrs

kk

(12)

We define the Curvature Vector

mmm PNormNorm

NormNormPU 4

2

2m

2

2i

ii

im P,,

)P(PP),P(P

PP),P(P

(13)

Suchard

which from [7] and simple calculations, should have the meaning mm BCa

U 221

such that

denotes a 4-acceleration field that will accelerate every particle, that can measure proper time, by an anti-symmetric matrix, and C is the speed of light and is a rotation matrix, i.e.

, is a vector and is the General Relativity metric tensor. The curvature itself does not depend on any specific acceleration since it is a scalar field.

ma

mB k

ki

im

m VVgVBVB

mV g

m

mUUCurv 412 (14)

Obviously

and therefore like 4-acceleration that is perpendicular to 4-velocity is

perpendicular to . In its complex form (13) becomes

0 PU

PU

22 * and Zand

2P*P*PP

ZZ

PPZZZU

dxdZN

kk

(15)

and by using )**(21 k

kk

k UUUU

))**(21(

412 k

kk

k UUUUCurv

(16)

Obviously . 0*

PU

Possible sources for an acceleration field: An acceleration field can be represented by the Tzvi

Scarr and Yaakov Friedman [7] matrix as 2

*

U

PPPA

kk

such that AA

We may write this matrix explicitly by (15) and (16) and also require an additional scaling and reach the following anti-symmetric singular matrix

2/1

2/2

***2

UZPUPPPU

ZPA

ZUPPU

A

So Z

PUUZ

PAA kk 4)*()**(

. It easily verifiable that 0)( ADet . What is required,

however, is that . 0)( ADet So we need a modified . On our path we will see symmetries that are usually achieved by second quantization and particle symmetries but with a very different geometric source.

A

9

Suchard

5. THE STANDARD MODEL SU(3) x SU(2) x U(1) SYMMETRIES VIA THE GODBILLON-VEY CLASS

Caution: Please note, the following is NOT a quantum theory of the discussed acceleration field . The sole purpose of the following equations, is to show the degrees of freedom in the matrix representation of the acceleration field action.

U

We now present the curvature quite closely to the Reinhart-Wood metric formula [2],

),(),(),(),()(2

1

))*

(*

(2

12 22

v

kk

kk

ZP

ZP

ZP

ZP

ZPZ

ZPZ

Z

ZPPPZ

ZPZ

ZPPPZ

ZPZ

ZZUPPU

A

(17)

For some conserving field . It is immediate that if is imaginaryso is because

then iscomplex,as

dxd / P U

A Z isreal.Thedivergenceoftherealpart isthen and this factwill become crucialwhen electrogravity is discussed. It is easily verifiable that the

exterior derivative

02/( UU );*

dxZ

Pdd )( , can be written with dx

U

2 by the Godbillon-Vey

observation [2][3], as dx^

)(FT

dxZ

PUdxdxZ

Pij 2

^),( d )^( with coordinates

. Another observation is the Godbillon-Vey class, on the foliations whose tangent bundle is

perpendicular to

x

ZP . )(3 MH

][

^^^2,

2)( ddxdxdxU

UFGV is closed on

. More important is the observation that is the Reeb cohomology Class )(FT ][ U . From Reeb it follows that the projection of the curvature vector on the tangent bundle of

the foliations that are perpendicular to , have a vanishing exterior derivative. In other words,

is a closed form on i.e.

U )(FTF P

))(FU F 0)(;(; FU F

FU . . From (6) and the remark after (13) it follows that the rotor of the electric field should be zero on , but that is possible only if even photons are pairs of charged particles, possibly oscillating with a total zero dipole moment as indeed informally predicted by physicists such as Hans W. Giertz [11]. This conclusion, as will be discussed, regards very important late findings by NASA. It is worth mentioning that the expansion of the holonomy of the

foliations is dictated by the norm of the Reeb vector [12] which in the real case is UU and therefore this value is of great cosmological interest. Following is an offered complex form of (17),

10

Suchard

)2

**(

41))**()**((

161

...))2

****)(2

((41)**(

41

UUUUUUUUUUUU

ZUPPU

ZUPPU

AAAA

(18)

which is the time field curvature action. In the real case where *AA we define AF 21

and the action (16) takes the form which reminds of the electro-magnetic theory but without the vector potential. As we shall see, this vector potential is indeed redundant because electric charge is a geometric phenomena. From the theory of Lie Algebras,

FF

exp()) 1)0(exp())(exp( ATraceADet

)4()exp( SUA A

and therefore either or

depending on whether is real or not. We proved that does

rotate and scale

)4()exp( SOA

k

k gAA

ZP * into . So by the Tzvi Scarr and Yaakov Friedman representation [7], for

every test particle with rest mass and with velocity and the speed of light we have,

2/U

V C

ddV

CCVA 2

1 (19)

Just like the term U

ddV

C 21

represents a curvature vector but of a specific particle related

trajectory. We continue with the Tzvi Scarr and Yaakov Friedman acceleration representation [7] matrix and for simplicity we restrict our discussion to the real case. is singular and we can easily define a

matrix that rotates vectors in a plane perpendicular to both and to . That is the matrix AU P

AB

21

(20)

Where is the Levi-Civita symbol. It is easily verified that

))(( BBAABABA

and also

AAAAAAAA

AAAABB

jiijji

jiij

jiij

)(21))((

21

21

21

Therefore

AABBAABABA

2))(( j

i is the Kronecker delta.

BAA (21)

11

Suchard

12

(21) is the matrix we have been looking for and it also results in an immediate degree of freedom in (16). In particular,

~ rotations in [13] that do not affect may be applied to . These

rotations are in ,

)4(SU

A B

)1(xUSU )2( ~~ BB and

~~

A A .

~~ BAA (22)

Diracs Monopole: If the reduction of tothefoliationtangentspace that isthekernelof

changesinrelationtothepoint intheaxissystemreducedto ,thenthedeterminantofkU )(FT

)kP

0 (FT

~~

A B preservesthesamesign.

For the real case, where , note that kk UU * 4/)()(41

2

BBAAUUCurv kk .

SU(3) degrees of freedom result from a way to represent Z

P with 3 scalar functions

according to the Frobenius (foliation) theorem. The Lie brackets of 3 vector fields on a

perpendicular foliation to

)3(),2(),1( qqq

ZP , have to depend on the tangent bundle of the foliation as

follows,

)(FT F

0))3(,)2(,)3()2(()3(,)2(,)1( 4)

0))3(,)1(,)3()1(()3(,)2(,)1( 3)

0))2(,)1(,)2()1(()3(,)2(,)1( 2)

0)3()2()1( 1)

).3()3(),2()2(),1()1(

complex. is 0 ,)3()2()1(

rrsrs

rkji

skji

rrsrs

rkji

skji

rrsrs

rkji

skji

kjikji

ssjj

jssj

qqqqqqq

qqqqqqq

qqqqqqq

qqq

qdxdqq

dxdqq

dxdq

Pqqq

(23)

Conditions 2,3,4 in (23) define the gradients of , and as Holonomic . )1(q )2(q )3(q

Vectors are Holonomic if their Lie brackets depend on them for some

coefficients . Condition 1 is a transversality condition. The Lie brackets of each two vectors must

depend on the vectors that span . (23) has a deep meaning that our scalar field of time is a result of 3 scalar fields of space locally perpendicular to the gradient of the scalar field of time.

)(sh

jc

3

1)()](),([

jj jhckhih

)(FT

The source of these fields are possibly the Sam Vaknin's Chronon field [1]. This implies a simpler physics and it is possible that Dirac's equation

[14] and spinors are an algebraic language that was required because concurrent physics theories do not have a complete analytic theory

Suchard

of space-time. The need for algebraic abstraction may not be related to physical reality but rather to the way we perceive it. This possibility justifies further extensive research and should not be dismissed. 5.1 Dr. Vaknin's Theory One of Four of

his Offers The coming definitions will reflect Dr. Sam Vaknin's view, that even photons by entanglement of wave functions have rest mass. Not that a photon as observed on its own has rest mass. That is incorrect. Matter by Vaknin's theory [1] is a result of interaction of a field of time that quite reminds of Quarks in which summation results in positive propagation. This paper sees Dr. Sam Vaknin's theory as a starting point. Dr. Sam Vaknin's possible description of time: "Time as a wave function with observer-mediated collapse. Entanglement of all Chronons at the exact "monent" of the Big Bang. A relativistic QFT with Chronons as Field Quanta (excited states.) The integration is achieved via the quantum superpositions". We will now refer to a simple implementation of Sam Vaknins approach as a quantization idea of

time, as the action

4

424

dgCurvK

C

Where g

is the root of the negative metric tensor determinant for the volume element, such that

AtomTimenPn

_))(...)2()1(lim(

and such that: 1)(*)(

4

4

dgkk

And 0)(*)(0

4

4

dgkjkj

Here are more definitions we will need. Energy Conservation: In any physical system and its interaction, the sum of kinetic (visible) and latent (dark) energy is constant, gain of energy is maximal and loss of energy is minimal. See E. E. Escultura [15].

Information: Information is a mathematical representation of the state of a physical system with as few labels as possible. Labels can be numbers or any other mathematical object.

Energy Density: We define 24

CurvK

C

, such

that is the speed of light and C K is the gravity constant, as the Energy Density of space-time. If this value is defined by (14) then P is the upper limit of measurable time from an event back to near big bang event or to a sub-manifold of events and therefore (14) is intrinsic to the space-time manifold because it is dictated by the equations of gravity and adds no information that is not included in the manifold and in the equations. As we shall see, if we choose to write

P such that is a complex scalar field then if is a function of only then (16) is reduced to (14) as if P . Consider the set of events for which is constant. Since is not a coordinate, we can't expect that set to be a sub-manifold but a unification of such 3 dimensional geometric objects . )0( )0 (

3 3We consider as a Morse function on the space-time M manifold. That is that M is locally smooth and the differential of this map is of rank 1. In such a case the Morse Sard theorem states that the Lebesgue measure of the Critical Points of M is zero [16]. Energy: The following is equivalent to rest mass energy. The integration of

)( 03

)(

24

03

dCurvK

C is defined as the

Energy of the scalar 0 . This value is locally conserved for small neighborhoods in if

as any local integration of the squared norm of a vector field is conserved if its divergence is zero. So if there is a possibility of

local conservation of the term Energy does not hold. Photons too, by entanglement and superposition have rest mass but not as an isolated electro-magnetic wave.

)( 03

0; mmU

0; mmU

nice thought experiment is the use of a covariant Gauss Law and the atom of space for

8 . After we have these definitions in mind, we may want to extend Gauss Law to a covariant format and later use the idea that there is an

13

Suchard

atomic structure of space time that can't be curved by gravity. For start, a generalized Gauss law using the exterior algebra, should look like:

033

^^^

QcTaudxdxdxEdsES

kji

S

(24) such that E is the electric field. a surface element and

dsx denotes the coordinates, but here

E is a 4-vector and will have to be defined, is the length of the path of a charge

enclosed in a 3 dimensional surface , cTau

3S 0 the permittivity of vacuum and is the speed of light. We also assume that in our local coordinates, the metric tensor is

c

,1 1,1,1 3322 1100 gg0

Tau

g

gg otherwise, , i.e. flat geometry on the hyper-cylinder. Since space-time can be curved, we can only discuss small volume and small proper time in local coordinates. Instead of considering the general acceleration field

3S

ddV

CCVA 2

1 as in (18), we need a

representative trajectory dependent boost 4-

acceleration

a

ddV

a

so now we can apply

(6) as an exact relation as the outcome of the energy density i.e. for the

previously discussed guess

)Ka 8/( 8 .

E

Ka

04

(25)

such that E is a 4-vector that replaces the electrostatic field. Please notice that here

. Consider a 3 dimensional surface of 4 dimensional cylinder around a charge Q , i.e.

around a

00 E

Surface xIS2 xICylinder 3B such that both the radius of the sphere and of

the ball and the length of the interval 2S

3B I , is L and is small. We may consider L to be an atom of length so by the relation we have from (18), acceleration is merely a curvature of a particle's trajectory. If there is an atom of length, say L , then the maximal curvature would appear in loops of radius L so it would be

LCurvature 1 and by (18) and by the relation

VAca

2 such that , the maximal

acceleration is therefore

1VV

LCaa

2

(26)

where is the speed of light. By (6), this acceleration can be the underlying acceleration field in an electric field. The parameterization in polar coordinates of the surface is

C

)xIS2),sin( l)sin(),cos( LL ),cos() Lcos((

where Ll 0 . It is )0),sin(sin(),cos()cos(( ),cos() rrr

0

where l)),sin(sin(),cos()cos(( Lrr

and it is ),cos() r

where Ll , such that )2/),2,0(),,0( ,2/( Lr

and the normal to the surface which for Ll 0

(cos( is

)0),sin(sin(),cos() ),cos() n and at Ll the normal vector is and at

)1,0,0,0(

0n

l the normal is . )1,0 ,0,0(n

A

nE

An acceleration field around a physical source is expected to cause

to cancel out

on the "top" and "bottom" and Ll 0l boundaries combined together, because acceleration changes sign as the time coordinate changes sign. So the integration (25) will be by (25) and by (26), even though the metric is Minkowsky,

02

00

23

4

44

KCQ

QLKL

CL

L

A

(27)

But what about gravity which results from the energy of the acceleration field and what about the influence of such gravity on the metric tensor within the cylinder ?

14

Suchard

15

Since we consider an atom of length L , no field can exist below that radius and therefore, (24) is valid even if the charge is not small. Q

Here we are about to explore another degree of freedom in the action operator of the acceleration field as shown by a representative vector field

idxdP

that curves.

We now assign the electron charge to Q and (27) becomes,

e Caution: Although the calculation may have a quantum meaning, it is not brought here for the purpose of developing a quantum field theory.

02 4

KCeL (28)

We revisit our acceleration field,

2/)**(

* 24

2

2

2

ii

ii

m

m

m

PPPP

s.t. NPN

P,NN

,NU

(also found

as Z in this paper) we can sloppily omit the comma for the sake of brevity the same way we

write instead of for iP iP, idxdP

and write

m

P

What is the wave length of a photon that can be emitted by the field E ? We know that the energy should be times smaller than the energy of the field, such that is known as the "Fine Structure Constant". Therefore, the wavelength is bigger by /1 than the expected wave length if we assume that L is also the wavelength of a photon that has equal energy to the field E or in its matrix form .

We have

A

mm N

PNN

N 2U 42

2

* .

0

2

K

4Ce

L

assign

Ce

2

041

and get,

Suppose that we replace P by such that is positive and increasing, then

f(P) f

ip(P)Pf

2

idxdP

f p

ii dppdf

dxdf(P)f(P) )( . Let

then

and

PP2 N

2 f(P)N 2 (P)Nf(P)

kpp

ppkk

pfpf

NN

NN

)()(2

2

2

2

2

but also

3CK

(29)

Which is known as the "Planck Length". 6. INVARIANCE UNDER DIFFERENT

FUNCTIONS OF P

kk

k

p

kps

ps

p

ppsk

p

ppk

kps

pskk

UPN

PNNN

pfNppfppf

ppfpf

NNp

pfpf

NN

N

ppfppf

NN

NNU

4

2

2

2

222

2

2

2

22

2

2

2

)()()(

))()(2

()()(2

)()(

(30)

Consider quantum coupling between the wave function of a particle and the time field ,

as follows. Where does this coupling ** 2PP P come from ? It is has some common sense if we say that the sum of wave functions that intersect/coincide with an event, influence the time

Suchard

measurement from near the big bang singularity event or from a sub-manifold of events to that specific event. A much better choice that will be discussed in the appendix "Appendix - Event Theory and Lvy Process" is P and ** PP . In that case, * is a Lvy measure and time itself becomes a stochastic process with events in which time can be measured. The appendix presents an offer of how to quantize (14). Currently, we defer the quantization of (14) and we define the curvature vector of P ,

k)

kx N ( 2

k N

NNNU (

)(*)(

22

jj

2

2

k2

(31)

Index k means derivative by coordinate , , .

kk *)()

kkN

2

As a special case, we replace by a wave function that depends on only

1-i s.t.

iE

e (32) E is the energy of a coupled particle, is the Barred Planck constant, so we have

)1()(

Eikkkk

(33)

)1()1( 222

22

222

ENEN kk

(34)

and

)(2

)1(

/2

)1(

)1(

222

2

22

22

222

2

22

2

22

2

2

2

2

NN

NENE

E

E

NN

NN ssss

2

2

EEs

(35)

Now we want to calculate kNN )(

)(*)(

22

jj

2

so we have

)(2

)()

)(2

(

))1()1(

))1(*()

)(2

(

)()(

*)(

222

2

22

2

2222

2

2

2

22

22222

2

2

2

22

jj

2

EE

NN

NEE

NN

Ei

NE

Ei

EE

NN

NN

kkj

jk

jjj

k

j

jj

k

(36)

From (31), (35) and (36) we have the result

16

Suchard

kk

jjkk

jjk

kkj

jkk

kk

UN

PPNNN

NN

NN

EE

NN

EE

NN

NN

NNU

))(

())(

(

)))(

2)(

())(

2((

)()(

*)(

22

2

2

2

22

2

2

2

222

2

22

2

222

2

2

2

22

jj

2

2

k2

(37)

7. GENERAL RELATIVITY FOR THE DETERMINISTIC LIMIT By General Relativity, We have to add the Hilbert action to the negative sign of the square curvature of the gradient of the time field. Negative means that the curvature operator is mostly negative. We assume 8 .

dgLRMinActionMin

RicciRZ

PPZZZUN

kk

.821

curvature.

UU41L and and PPZ k

k2

2

(38)

A reader that still insists on asking on where does come from, can understand that L can be developed also for and remain invariant if is only a smooth function of . If P then

)U*U k*U(U81

kkk L and an integration constraint can be

1)(*

)(

33

dg (39) Caution: is not a sub-manifold because )(3 is not a local coordinate and thus the local submersion theorem [17], [18] does not hold. However, is necessarily a countable unification of three dimensional sub-manifolds - Almost Everywhere - on which

)(3 is stationary due to

dimensionality considerations. R is the Ricci curvature [19], [20] and g is the determinant of the metric tensor used for the 4-volume element as in tensor densities [21]. Important: If instead of P we choose P then in (37) is the same if kU depends only on . Moreover, instead of the constraint in (39) the following

1*)(

44

dg (40)

17

Suchard

leads to a theory in which represents an event and EventdgPP )(4

4* represents the

time of an event, i.e. see "Appendix Event Theory and Lvy Process". Now we return to (38). By Euler Lagrange,

g

ZPP

Z)Z(Pg

Z)Z(P

P)ZZ

P),P(P(ZPP

Z)Z(PPP);P

ZP),P(P(

g

gZ

)Z(PP)P

Z)P),P((P

(P)ZZ

P),P(P(

)PPPPPP(Z

P),P(PP);PP

ZP),P(P

(

)PPPPPP(Z

P),P(P);PPP

ZP),P(P

(

),(g)g(L

dxd

g)g(L

PP s.t. ZZ

)Z(PL

ss

kk

mm

ss

ss

mim

i

mim

i

ss

m

m

ss

mim

i

mim

i

ss

mm

ss

mm

3

2

3

2

33

2

3

3

2

4

2

3

33

33

3

2

21

222

2132

22

22

(41)

18

Suchard

g)Z

ZZg

ZZZ

ZPP

ZZZPP);

ZZ((

g

g)P(P

ZZPPZ

ZZZ

ZZ

)Z

)ZPPZPP(Z

)Z);P(P

)Z

)ZPPZPP();

ZPPZ

(

,g)g(L

dxd

g)g(L

PP s.t. ZZ

ZZL

kk

m

m

i

i

mm

ss

mim

i

mim

i

mm

mim

i

mim

i

m

m

mm

2222

232

22

22

2

2122

212

22

22

g)Z

ZZg

ZZZ

ZPP

ZZZPP);

ZZ((

g

g)P(P

ZZPPZ

ZZZ

ZZ

)Z

)ZPPZPP(Z

)Z);P(P

)Z

)ZPPZPP();

ZPPZ

(

,g)g(L

dxd

g)g(L

PP s.t. ZZ

ZZL

kk

m

m

i

i

mm

ss

mim

i

mim

i

mm

mim

i

mim

i

m

m

mm

2222

232

22

22

2

2122

212

22

22 (42)

19

Suchard

g)ZPP

;UgUUU(U

g.

gUUUU

ZPP

ZZZ

ZPP

Z)Z(P

P)P);ZZ();P

ZP),P(P((

g.

ZPP

Z)Z(PP)Z

ZP),P(P(

ZZZ

gZ

ZZgZ

)Z(PZPP

ZZZ

ZPP

Z)Z(P

P)P);ZZ();P

ZP),P(P((

g.

)Z

ZZg

ZZZ

ZPP

ZZZPP);

ZZ((

ZPP

Z)Z(Pg

Z)Z(P

P)ZZ

P),P(P(ZPP

Z)Z(P

PP);PZ

P),P(P(

,g)g(L

dxd

g)g(L

UUand LZ

PPZZZ and UPPZ

k

k

kk

k

k

m

m

kk

mm

ss

kk

m

m

kk

mm

kk

m

m

ss

kk

mm

mm

k

kk

221

21

22

22

2

21

21

22

22

2122

21

22

2

23

2

23

3

2

32

23

2

23

2

23

2222

3

2

3

2

33

2

3

2

(43)

20

Suchard

g)PZ

)P(ZZ

ZPZP);Z

)PP(Z((

g

PZ

)P(ZZ

ZPZ

PPZ

)P(ZP;PZ

)P(Z

PPZ

)P(Z);PPZ

)P(Z

,P)g(L

dxd

P)g(L

),P(P and ZPP s.t. ZZ

)P(ZL

m

mm

m

ss

m

mm

m

kik

i

ss

ss

i

i

ss

ss

m

ms

s

4

2

33

4

2

3

33

33

3

2

624

62

44

4(4

(44)

g)PZ

ZZ);ZZ((

g

gPZ

ZZ

ZPZ

Z;PZ

ZPZ

);Z

ZP(

,P)g(L

dxd

P)g(L

),P(P and ZPP s.t. ZZ

ZZL

m

m

m

m

kik

i

kik

i

m

ms

s

32

3

22

22

2

44

4

44

44 (45)

From (38), (43), (44) and (45), we get two tensor equations of gravity, assuming 8 , where the metric variation equations (38) and (43) yield,

21

Suchard

pkjpjk

Ppj

Pkpp

Pkjjk

kkk

k

m

m

kk

m

ii

kk

R

gRR

RgRZPP

UgUUUU

gUUUU

ZPP

ZZZ

ZPP

ZZP

PPZZP

ZP

UULZ

PPZZZUN

k

k

23

2

23m

kk

22

),(),( s.t.

s.t.21);2

21(

418

.

21

2)(

2

));(2);),P(P

((2

418

PP Zand 41 , , PPZ

(46)

R is the Ricci tensor. In general by (7) (46) can be written as

RgR

ZPP

UgUUUU kkk

21);2

21(8

41

k

We can also see that the ordinary local conservation laws are modified if unless the local

average around charge

0; kkU

0);;2( ZPP

U kk which is expected due to symmetry around charge.

Different charges will fall at the same speed, for a detailed proof of conservation see APPENDIX - Conservation, Why Do Different Charges Fall At The Same Speed.

Majorana - like field: The term ZPP

;U kk2 in (46) can be generalized to:

ZPPPP

;U;U kk

kk 2/)**()2/)*((2

Which is an interesting case because if the field is fully imaginary, P PP * or or

then which means that there is no electric charge. It could be the case in

Neutrinos. In that case, since the acceleration matrix is

0);( ZZ

0Z 0* kk

kk ;U;U

),(Z

P

),( Z

PA it is possible that the

norm of i.e. P Z is imaginary which means 0Z and is space-like. The complimentary

matrix

kP

A

21B can be transformed to a real matrix due to the SU(2) x U(1) degrees of

freedom and also be imaginary.

22

Suchard

Important: The force represented by may greatly differ from what conventional physics defines as mutual interaction forces. This is because such a force is a property of space-time (although not anticipated from the metric alone) and as such, it can be construed as an interaction with space-time itself, i.e. all matter. Please note that is a curvature vector which means time can't be measured along geodesic curves as these curves are prohibited in the field (17),(18).

U

U

Denoting LL dxdZZ , from the vanishing of the divergence of Einstein-Grossmanns tensor we have:

kkk

kk

k

LLk

kk

kk

k

LL

k

LLk

kk

kk

k

LL

k

LLk

kk

kk

kkk

kk

kk

kkks

kks

s

kkk

RgRZPPUP

ZPZUUU

ZPPUP

ZPZP

ZPZUUU

ZPPUP

ZPZP

ZPZ

ZZ

ZZU

UUZPPUUUUUUU

ZPPUggUUggUU

UUUUZPPUgUUUU

);21(40);;2(),(;

);;2());(),((;

);;2());();();();((

;

);;2(;;;

);;2();;(21

;;

);;221(

2

22

22

k

sk

k

(47)

The term PZ

PZU kL

Lk ),( 2 is due to the fact that k

LL

kk P

ZPZ

ZZU 2 and from

which

0kk PU

0));(( 2 kL

Lk PZ

PZU . The bottom line is that it is possible that

0);;2(

ZPPU k

k , though by local averaging, due to symmetry, it is expected that

0);;2( k

kk

ZPPU . We will later refer to -

ZPPU k

k

;2 as electro-gravity.

We can now explore the relation between local parallel translation in loops and non-zero divergence

. Contraction of (46) by 0; kkU

ZPP

yields

23

Suchard

RZPPRUUU

ZPPRgR

ZPP

ZPP

UgUUUU

kkk

kkk

21);2

21(8

41

)21();2

21(8

41

k

k

(48)

Contraction of (46) by yields g

RUUU kkk

21);

21(8

41

k

(49)

So from both equations we have,

ZPPRU k

k

;841

(50)

If the divergence is related to electric charge then this result implies that even photons generate pairs of oscillating charge. By the definition of the Ricci tensor

kkU ;

ZPPP

ZPPR

ZPPRRR

LL

LL

LL

LL

);;;;(

(51)

This is because by the definition of the Riemann curvature tensor, the anti-symmetric derivation of a vector field yields and so LV VRVV jLjk

Lkj

Lk ;;;;

k

jLk

jkL

kjL VVRVVV k );;;;(

kL

LkLk

LkL

L VVRVVV k );;;;( and therefore

kVVR k

ZPPPU

ZPPRU

LL

LL

kk

kk

);;;;(;841

;841

(52)

Definition: We will call the latter "Charge Holonomy Equation". Or in its complex form,

)*);;;;Re(();*;(881

ZPPPUU L

LL

Lk

kk

k

(53)

24

Suchard

And if 8 we have ZPPPU L

LL

Lk

k

);;;;(;41

in the real case and

)*);;;;Re(();*;(81

ZPPPUU L

LL

Lk

kk

k

in the complex case.

For Z

PPP LL

LL

*);;;;( not to be zero, it requires that locally, parallel translation along

different paths that connect two events, will yield different results. Obviously because P is a scalar field, in flat geometry and in curved geometry we can have

.

0);;;;( Lk

Lk PP

0);;;;( Lk

Lk PP

By Lee C. Loveridge [22], another illuminating equation which is simple if 8 is,

);41(8)3( k

kUUUR

(54)

or in the complex case );*;)**(41(

218)3( k

kk

k UUUUUUR

Where is a three dimensions scalar curvature in the space perpendicular to or in the

complex case, to

)3(R ,PP 2/)*( PP . (54) can be viewed as a non linear version of Proportional to

square error Differential case of PID control which is well known to physicists who also have

background in engineering. UU

41

is the square proportional term and is the differential

term. The three dimensional scalar curvature is merely an output of such a control system. Note

that both and

kkU ;

)3(R

)3(R

P

UU41

are curvature terms where describes the curvature of the gradient

of the time field i.e. the curvature of .

U

From (44),(45) we have,

0;))(,

(

gWgUU

Pdxd

Pdxd k

k

We recall, ))(,

( gUUPdx

dP

W kk

25

Suchard

02));((4

)(2

)(4);)((4

P4P);(4

)(62);)((4P)4);(4(

2m

22m

4

2

3

3m

2

4

2

3m

33m

2

UZ

PZPZUU

ZU

ZPPZ

ZZ

ZPZ

PZPZP

ZPPZ

ZZZ

ZZ

PZPZ

ZZPZP

ZPPZ

ZZZ

ZZ

W

mk

k

k

k

mm

m

mm

ss

m

mm

mss

m

0;)(

2;4; 2

U

ZPZ

ZPUW

mm (55)

A simpler solution to zero Euler Lagrange equations, is

0))(,

(

gUU

Pdxd

Pk

k

(56)

Which results in a special case, Zero Charge as charged particles are related to non-zero divergences and either or . 0

UU 0mm PZ

0);( U (57)

and (46) becomes,

RgRgUUUU k

21)

21(

418

k (58)

The reader can either refer to the following calculation or skip it.

26

Suchard

02));((4

)(2

)(4);

)((4

P4P);(4

)(62

);)(

(4P)4);(4(

2m

22m

4

2

3

3m

2

4

2

3m

33m

2

UZ

PZPZUU

ZU

ZPPZ

ZZ

ZPZ

PZPZP

ZPPZ

ZZZ

ZZ

PZPZ

ZZPZ

PZ

PPZZ

ZZZZ

mk

k

k

k

mm

m

mm

ss

m

mm

m

ss

m

(59)

Recall that , multiplication by 0kk PU

4-P and contraction yields,

0)(

));)(

();(( 32

2m

32 ZPZ

ZZZZ

ZPPZ

ZZ mm

mss

(60)

0))(

(1);)(

();( 32

2m

32 ZPZ

ZZZ

ZZPPZ

ZZ mm

mss

(61)

and as a result of (61), the following term from (46) vanishes,

0))(

(1);),P(P

();(2

2);(2);(2

3

2s

23m

2

s

kk

m

m

m

kk

k

k

kk

PPZPZ

ZZZ

ZP

ZP

ZZ

ZPPUUPP

ZU

ZPPU

(62)

which yields a simpler equation (58). Recall that 2)(

ZPPZ

ZZU

ss

,

And that UUU

ZZ

0);(1

)(11);(1)(1);(

))(

(1);)(

();(

m2m

3

2

2m

32

UZ

UUZ

ZUZ

UZ

UUZZ

UZPZ

ZZZ

ZZPPZ

ZZ

mm

mm

mss

(63)

which proves (57).

27

Suchard

Question to the reader: If ZPP

UgUUUU kkk

;221

k describes the electro-magnetic

energy momentum tensor, where is the torsion tensor FF that is so basic to the electro-magnetic theory? Answer: is not any electro-magnetic field. It is a property of space-time as U P is dictated by the equations of gravity. , however, offers a way to describe an anti-symmetric tensor which is a singular Lie Algebra matrix as an acceleration field via Tzvi Scarr and Yaakov Friedman

representation [7],

U

2/)**(*

2

A

PPPPPU

AA such that . See "Appendix

Acceleration field representation". The field is actually represented by but it is not an electromagnetic field, rather, it is the underlying mechanism that results in what we call, the electric

field. In the complex formalism either,

A

*)2/)*

*

PP*( PP

P

(2*

AU or

2/)**2

PPPAU

(*

PP. Increasing or decreasing )*

UUK

C *(814

U

A

U results

in change of Energy density and in the phenomena we call Electro-Magnetism. represents a non-inertial acceleration of every particle that can measure proper time and not of photons as single particles.

Inertia Tensor: We define inertia tensor as

gUUUU kk21

418

this reflects what we know

as energy momentum tensor.

Question: why is the tensor k

kkk

PP

PPU ;

42

not included in the Inertia Tensor ?

Answer: Because the term k

k PP

PP is not a material field.

Electro gravity tensor: We define the electro-gravity tensor as k

kkk

PP

PPU ;

42

.

As we shall see, kkU ;

21

is equivalent to electric charge density 20

4;21

CKU k

k

or by (7)

202

;21

CKU k

k

if 8 .

But the sign could be also plus. Charge conservation yields the following:

From, kkk

kkk UUUg

ZPP

UgUUUU ;2);221( kk

it follows

28

Suchard

29

that

dgRdgUUdgUUU kkkk

kk 41);2(

41

.

Construction and Destruction: We define construction or destruction as local appearance and

disappearance of non zero kkU ;

21

neighborhoods of space-time as a function of . This definition

alludes to the well known terms of construction and destruction brackets in Quantum Mechanics. 8. RESULTS - ELECTRO-GRAVITATIONAL THRUST, DARK MATTER AND DARK

ENERGY Dark Matter: Dark Matter will be defined as additional Gravity not due to the Inertia Tensor. It is meant that the cause of such gravity is not inertial mass that resists non-inertial acceleration. It also emanates from the acceleration field as expressed by the Scarr-Friedman matrix [7], AA . This field prohibits Minkowsky geodesic motion of rest mass. Dark Energy: Dark Energy will be defined as negative Gravity not due to the Inertia Tensor. It is meant that the cause of such gravity is not inertial mass that resists non-inertial acceleration. Also in this case electro-gravity is not the only cause. The acceleration field that prohibits Minkowsky geodesic motion of rest mass must also be taken into account. The following will describe a technology that can take energy from space-time apparently by Sciama Inertial Induction [23] and is closely related to Alcubierre Warp Drive [24]. Electro-gravity follows from (6), (46) and (55). For several reasons we may assume the weak acceleration of uncharged particles mentioned in (6) is from positive to negative charge see also [8], consider the general relativity equation

AA

RgRGZPP

UgUUUU kkk

21);2

21(

41

k such that the Ricci tensor is

pkjpjkP

pjP

kppP

kjjkR

),(),(

G is the Einstein-Grossmanns tensor. From (4) in a weak gravitational background field and 8 ,

2022 4)(21

21

CEK

CaP

)P(PP),P(P

PP),P(PU mm

ii

ii

m

m (64)

and also

20

2 2 CEK

Ca m

0;0 kE (66)

C is the speed of light, is the non-relativistic weak acceleration of an uncharged particle,

a0

is the permittivity constant in vacuum, K is the gravitational constant and E is a static non-relativistic electric field in weak gravity, assuming that by correct choice of coordinates,

The reader is requested to notice that (70) is only a non-relativistic and non-covariant limit! From electro-magnetism

0

;

kkE (67)

),E,E,E(EEm 3210 0 (65) Such that is the charge density

Suchard

)21(1

28

)21(

418

k40

k

gEEEEC

K

gUUUU

k

k

(68)

Or by (7) if 8 (68) becomes

ZPP

CEK

ZPP

U kk

kk

20 ;

2;2

41

(69)

From the electro-magnetic theory 0

;

kkE

such that is the charge density and so for Schwarzschild coordinate time,

t

00202

GGC

Ktt

(70)

So 002

04 2

1 GGCKC tt

such that

K021

behaves like mass density and

therefore we can define an electro-gravitational virtual mass as dependent on charge : Q

KQQ

KMVirtual

00 221

(71)

We will calculate K

QMassVirtual02

_

for 20 Coulombs by assuming 8 ,

KgK

Coulomb 90

10417128083379310603588885.8023316916 1

.

Multiplied by 20 we have

.10683625616675882120717771.1604663316 20 11

0

KgK

Coulombs

Within 1 cubic meter the effect would be a feasible electro-gravitational field because Newton's gravitational acceleration as a rough approximation yields,

2

211

2

SecondMeter 105963166601003439588087.74477595

1/10683625616675882120717771.16046633

_

KgKradius

MassVirtualK

,

Consider parallel metal plates of 10cm x 10cm with a gap of 2cm and with low relative permittivity thin slab of 10 grams in the middle, such that the voltage of 37 Kilovolts is applied to the plates. The net classical non-relativistic gravitational force on the slab without regarding the non-geodesic acceleration field, is less than one third of a micro-Newton. That is a dauntingly small force which is very difficult to measure.

The calculations rule out any measurable vacuum thrust of Pico-Farad or less, asymmetrical capacitors even with 50000 volts supply, simply because the net effect depends on the total amount of separated charge which is

far from sufficient in standard Biefeld Brown capacitors [25]. Hypothetical use of sub-luminal photons: It is yet needed to be demonstrated that sub-luminal photons do exist and that if very close to the

30

Suchard

speed of light can interact with electric currents and/or fields as if they are high density electric charge distributions. After (17) there is a discussion on the vanishing of the Reeb class based on [2][3] from which the restriction of

to the foliation must have a zero rotor i.e. U

F(; 0)U)(;U FF

24101600

. This results in electric field which is always a result of electric charge, i.e. non vanishing divergence. The idea is that even photons can appear as a pair of oscillating negative and positive charge. Slight imbalance in charge distribution in photons can result in alleged effects such as EMDrive [26]. A slight 1% imbalance in +e and e charge of say

photons is equivalent to over separated Coulombs and therefore if

Hans Giertz assumption [11] holds, using photons for warp drive effects is the most feasible technological solution.

The 2_

radiusMassVirtualK

shows a gravitational

acceleration of over 126 gees between 1600 coulombs separated by a gap of 2

meters. In such a case, if a photon behaves like a dynamic oscillating or rotating dipole, these dipoles will be of different dipole moments aligned or anti-aligned with relativistic electric fields. The result is equivalent to true charge separation and can have a great technological potential for the development of a feasible warp drive thrust.

Use of plasma: Another idea is to use ionized plasma. Let us see what we can do with one gram of ionized hydrogen. The number of atoms by Avogadro's number is . The charge of the electron is so

and

2310 6.02214129n Coloumbs 1051.60217656 -19

-1-13 seckgme

-1110Coloumbs 105956868859.64853364 4Q 6.67384KF/m 10 76..8.85418781 -120

ssVirtual_Ma 5.59839925M Earth 5.97219

so 1 gram of hydrogen reaches a virtual mass of

. That is far less than the

mass of the Earth but the distance between two clouds of positive and negative ionized hydrogen can be much less than the average Earth radius and therefore a field that overcomes the Earth gravitational field is feasible.

Kg141081404220844444619768891 Kg2410

Dark Matter and Dark Energy follow immediately from negatively ionized gas in the galaxy and

positively ionized gas outside or on the outskirts if we assume energy density Kaa

8which means our

electro-gravity constant is KConst 8 . In other theories, e.g. KConst , the center of the galaxy should be positive. Negative charge as in the first case Const K8 , will only behave as Dark Matter more at distance because close to stars it is expected to cause induced dipoles within the star just as a small negatively charged ball above the Earth will polarize the ground. We now consider the classical non-covariant limit of the summation of two effects, the non-inertial acceleration and electro-gravity. Let be the charge of a ball at radius Q r , then by (7) the observed non-relativistic classical acceleration of an uncharged particle without any induced diploes is a

agravity _g ElectrorQK

rQ

KrKQa 2

02

002

)411(

24/2

K 02

If 8 then agrQKa gravityElectro _2

0

)41

81(

2

A friend, Mr. Yossi Avni suggested that two marble balls will be suspended on a balance above charged balls, left minus and right plus, and that by the checking if the balanced is tipped or not, we can decide 4 for unbiased balance and different value else, e.g. 8 . There is a problem, however, that charge in the ground below the balls will polarize the ground.

31

Suchard

Fig. 1. Avni suggestion, the left lower ball is negatively charged and the right is positively charged. The balls above are marble or even of lower dielectric constant

This theory means that a positive charge manifests attracting gravity but has a repelling acceleration field that acts even on uncharged particles that can measure proper time, i.e. have rest mass. The curvature and for positive charge . 0k

kUU 0; kkU

Negative charge manifests a repelling anti-gravity but has an acceleration field that attracts even uncharged particles and acts on particles that can measure time, i.e. have rest mass. The following table describes the relation between and the Dark Matter and Dark Energy.

Table 1. The relation between constants and Dark Matter and classical non-relativistic acceleration

Cause for dark matter

Cause for dark energy

Dark matter causes

0)411(

2

4

20

rQK

Positive charge Negative charge Caused by gravity

0)411(

2

4

20

rQK

Negative Charge

Positive charge Caused only by an Alcubierre Warp Drive [24] by induced dipoles. Positive side faces the negative galaxy center.

0)411(

2

4

20

rQK

Negative Charge Positive charge Caused by the acceleration field on particles with rest mass

9. CONCLUSION An upper limit on measurable time from each event backwards to the "big bang" singularity as a limit or from a manifold of events as in de Sitter or anti - de Sitter, may exist only as a limit and is not a practical physical observable in the usual

sense. Since more than one curve on which such time can be virtually measured, intersects the same event - as is the case in material fields which prohibit inertial motion, i.e. prohibit free fall - such time can't be realized as a coordinate. Nevertheless using such time as a scalar field, enables to describe matter as acceleration fields

32

Suchard

and it allows new physics to emerge as a replacement of the stress-energy-momentum tensor. The punch line is electro-gravity as a neat explanation of the Dark Matter effect and the advent of Sciama's Inertial Induction, which becomes realizable by separation of high electric charge. This paper totally rules out any measurable Biefeld Brown effect in vacuum on Pico-Farad or less, Ionocrafts due to insufficient amount of electric charge [25]. The electro-gravitational effect is due to field divergence and not directly due to intensity or gradient of the square norm. Inertial motion prohibition by material fields, e.g. intense electrostatic field, can be measured as a very small mass dependent force on neutral particles that have rest mass and thus can measure proper time. Such acceleration should be measured in very low capacitance capacitors in order to avoid electro-gravitational effect. The non-gravitational acceleration should be from the positive to the negative charge. The electro-gravitational effect is opposite in direction, requires large amounts of separated charge carriers and acts on the entire negative to positive dipole. ACKNOWLEDGEMENTS First, I should thank Dr. Sam Vaknin for his full support and for his ground breaking dissertation from 1982. Also thanks to professor Larry (Laurence) Horowitz from Tel Aviv university for his guidelines on an Early work. His letter from Wed, Jul 23, 2008, encouraged me to continue my research which started in 2003 and led to this paper. Special thanks to professor David Lovelock for his online help in understanding the calculus of variations and in fixing an error several years ago. Thanks to electrical engineers, Elad Dayan, Ran Timar and Benny Versano for their help in understanding how profoundly difficult it is to separate charge as required by the idea of electro-gravity. Thanks to other involved engineers, Arye Aldema, Erez Magen and to Dr. Lior Haviv from the Weizmann Institute of Science. Thanks to Zeev Jabotinsky and to Erez Ohana for his initiative of building a miniature warp drive engine with almost zero funds. Thanks to Mr. Yossi Avni who approved my research in 2003 into minimum cost diffeomorphism as part of a pattern matching algorithm. This early research helped in acquiring special knowledge in geometry that was later used in the presented theory.

Also a historical justice with the philosopher Rabbi Joseph Albo, Circa 1380-1444 must be made. In his book of principles, essay 18 appears to be the first known historical account of what Measurable Time In Hebrew "Zman Meshoar" and Immeasurable Time "Zman Bilti Meshoar are. His Idea of the immeasurable time as a limit [27], is the very reason for an 11 years of research and for this paper. COMPETING INTERESTS Author has declared that no competing interests exist. REFERENCES 1. S. Vaknin, Time Asymmetry

Revisited[microform], 1982, Library of Congress Doctorate Dissertation, Microfilm 85/871 (Q).

2. B. Reinhart and J. Wood. A Metric Formula for the GodbillonVey Invariant for Foliations. Proceedings of the American Mathematical Society 38: 427-430, 1973.

3. S. Hurder and R. Langevin. Dynamics and the Godbillon-Vey Class of C1 Foliations. preprint, September 2000.

4. E. H. Suchard, Electro-gravitational Technology via Chronon Field, Physical Science International Journal, pp 1158-1190, June/2014, 10.9734/PSIJ/2014/9676 Follow ups with minor constants update are available: Follow up 1 with constants update: http://www.scribd.com/doc/213903929/Electrogravity-Engine-Theory-and-Practice Follow up 2 with constants update: http://vixra.org/pdf/1407.0114v6.pdf Follow up 3 with constants update: http://freepdfhosting.com/f7a881b129.pdf In these follow ups, 10.9734/PSIJ/2014/9676 is correctly cited.

5. John W. Milnor, Topology from the Differentiable Viewpoint, pages 32-41, ISBN 0-691-04833-9

6. Hormander Lars, "The anaysis of linear partial differential operators I", 1983, Grundl. Math. Wissenschaft. 256, Springer, ISBN 3-540-12104-8 Chapter I. Test Functions pages 5-31, II 2.3 Distributions with compact support, page 44, Chapter VI. Composition with Smooth Maps, 6.3. Distributions on a Manifold.

7. Y. Friedman and T. Scarr, "Making the relativistic dynamics equation covariant: explicit solutions for motion under a

33

http://www.scribd.com/doc/213903929/Electrogravity-Engine-Theory-and-Practicehttp://www.scribd.com/doc/213903929/Electrogravity-Engine-Theory-and-Practicehttp://vixra.org/pdf/1407.0114v6.pdfhttp://freepdfhosting.com/f7a881b129.pdf

Suchard

34

constant force", Volume 86 Number 6, 2012 Physica Scripta 065008 doi:10.1088/0031-8949/86/06/065008

8. T. Datta, M. Yin, A. Dimofte, M. C. Bleiweiss

and Z. Cai, "Experimental

Indications of Electro-Gravity", 8 Sep 2005 arXiv:physics/0509068[physics.gen-ph]

9. Hector Luis Serrano, (President of Gravitec Inc.) capacitor at 1.72 x 10^-6 Torr, Experiment by NASA and Gravitec, Video from July 3, 2003.

10. Y. Doi, T. Makino, H. Kato et al. D. Takeuchi, "Deterministic Electrical Charge-State Initialization of Single Nitrogen-Vacancy Center in Diamond", Published 31 March 2014, Physical Review X 4, 011057, DOI: 10.1103/PhysRevX.4.011057, Publisher: American Physical Society.

11. Informal paper: Hans W. Giertz, "The Photon consists of a Positive and a Negative Charge, Measuring Gravity Waves reveals the Nature of Photons" http://vixra.org/pdf/1302.0127v1.pdf

12. G. Reeb. Sur certaines proprits topologiques des varietes feuilletees. Actualite Sci. Indust. 1183, Hermann, Paris (1952).

13. Walter Pfeifer, The Lie Algebra su(N), an Introduction , Chapter 5, The Lie Algebra SU(4), pp 87-106, 2003 (revised 2008), ISBN 3-7643-2418-X

14. Elie Cartan, The Theory of Spinors, Dover Publications Inc. Mineola, N.Y., ISBN 0-486-64070-1, p. 149 (14)

15. E. E. Escultura, Chaos, Turbulence and Fractal: Theory and Applications, International Journal of Modern Nonlinear Theory (SCIRP), pp. 176 185; http://www.scirp.org/journal/PaperInformation.aspx?PaperID=36849; doi: 10.4236/ijmnta.2013.23025.

16. Victor Guillemin / Alan Pollack, Differential Topology, Prentice-Hall, Englewood Cliffs, N.J.,1974. pp. 39-48, Sard's Theorem and Morse Functions, ISBN 0-13-212605-2

17. Victor Guillemin / Alan Pollack, Differential Topology, Submersions, Page 20, ISBN 0-13-212605-2

18. Victor Guillemin / Alan Pollack, Differential Topology, Submersions, Page 21,

Preimage Theorem, ISBN 0-13-212605-2

19. D. Lovelock and H. Rund, Tensors, Differential Forms and Variational Principles, Dover Publications Inc. Mineola, N.Y., ISBN 0-486-65840-6, p. 262, 3.27

20. D. Lovelock and H. Rund, Tensors, Differential Forms and Variational Principles, Dover Publications Inc. Mineola, N.Y., ISBN 0-486-65840-6, p. 261, 3.26

21. D. Lovelock and H. Rund, The Numerical Relative Tensors, Tensors, Differential Forms and Variational Principles, 4.2, Dover Publications Inc. Mineola, N.Y. , ISBN 0-486-65840-6, p. 113, 2.18, p. 114, 2.30

22. Lee C. Loveridge, "Physical and Geometric Interpretations of the Riemann Tensor, Ricci Tensor, and Scalar Curvature" DOI: 10.1103/PhysRevD.69.124009, Submitted on 23 Jan 2004, 7. The Einstein Tensor, (9), arxiv.org/abs/gr-qc/0401099

23. D.W. Sciama, "On the Origin of Inertia", Monthly Notices of the Royal Astronomical Society, Vol. 113, p.34, August 1952

24. M. Alcubierre, "The warp drive: hyper-fast travel within general relativity", Class.Quant.Grav.11:L73-L77, 1994, DOI: 10.1088/0264-9381/11/5/001

25. Thomas B. Bahder and Chris Fazi, "Force on an Asymmetric Capacitor", Army Research Laboratory ARL, March 2003, site: http://arxiv.org/ftp/physics/papers/0211/0211001.pdf

26. Matt Porter at IGN News, 28/April/2015, http://www.ign.com/articles/2015/04/28/nasa-may-have-invented-a-warp-drive

27. J. Albo, Book of Principles (Sefer Ha-ikarim), Immeasurable time - Ma